Dynamics analysis and numerical simulations of a stochastic delayed epidemic model with double disease driven by Lévy noise

Xueqing Li; Yingjia Guo; Xueqing Li; Yingjia Guo

doi:10.3934/math.2026122

AIMS Mathematics

2026, Volume 11, Issue 1: 3038-3095. doi: 10.3934/math.2026122

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Research article

Dynamics analysis and numerical simulations of a stochastic delayed epidemic model with double disease driven by Lévy noise

Xueqing Li ,
Yingjia Guo ^,

School of Mathematics and Statistics, Beihua University, Jilin, Jilin 132000, China

Received: 18 November 2025 Revised: 07 January 2026 Accepted: 20 January 2026 Published: 30 January 2026
MSC : 60G40, 60G51, 60H10

This paper investigates a stochastic delayed $SI_{1}I_{2}$ epidemic model with double epidemic and saturated incidence rate driven by Lévy noise. First, we used the stopping time theory to establish sufficient conditions for the existence of a unique positive solution. Furthermore, by the Lyapunov method and stochastic differential equation theory, we analyze the asymptotic properties of the stochastic delayed system around each equilibrium point. In addition, we show that both Lévy noise and time delay can affect the dynamics of the epidemic system. Finally, we use the Euler–Maruyama method to discretize the equations and perform numerical simulations to illustrate the theoretical results.
- Lévy noise,
- stochastic differential equation,
- delayed equation,
- Lyapunov function,
- existence and uniqueness,
- asymptotic behavior
Citation: Xueqing Li, Yingjia Guo. Dynamics analysis and numerical simulations of a stochastic delayed epidemic model with double disease driven by Lévy noise[J]. AIMS Mathematics, 2026, 11(1): 3038-3095. doi: 10.3934/math.2026122

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Abstract

This paper investigates a stochastic delayed $SI_{1}I_{2}$ epidemic model with double epidemic and saturated incidence rate driven by Lévy noise. First, we used the stopping time theory to establish sufficient conditions for the existence of a unique positive solution. Furthermore, by the Lyapunov method and stochastic differential equation theory, we analyze the asymptotic properties of the stochastic delayed system around each equilibrium point. In addition, we show that both Lévy noise and time delay can affect the dynamics of the epidemic system. Finally, we use the Euler–Maruyama method to discretize the equations and perform numerical simulations to illustrate the theoretical results.

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